Certain Bernstein-type \(L_p\) inequalities for polynomials

Let P(z) be a polynomial of degree n, then it is known that for \(\alpha \in {\mathbb {C}}\) with \(|\alpha |\le \frac{n}{2},\)

$$\begin{aligned} \underset{|z|=1}{\max }|\left| zP^{\prime }(z)-\alpha P(z)\right| \le \left| n-\alpha \right| \underset{|z|=1}{\max }|P(z)|. \end{aligned}$$

This inequality includes Bernstein’s inequality, concerning the estimate for \(|P^\prime (z)|\) over \(|z|\le 1,\) as a special case. In this paper, we extend this inequality to \(L_p\) norm which among other things shows that the condition on \(\alpha \) can be relaxed. We also prove similar inequalities for polynomials with restricted zeros.